Unanswered 'characteristic-p+characteristic-p+or+abelian-varieties' Questions

25 votes

0 answers

1k views

Status of the Euler characteristic in characteristic p

In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes: Enfin signalons que la situation en caractéristique positive est loin d'être aussi ...

Vivek Shende

8,723

asked Jul 7, 2013 at 7:32

20 votes

0 answers

408 views

Ado's theorem and the reduction to positive characteristic

The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case? The ...

Dmitrii Korshunov

2,934

asked Jul 31, 2023 at 2:53

17 votes

0 answers

1k views

Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties. Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]...

John Pardon

18.7k

asked Dec 28, 2013 at 19:48

15 votes

0 answers

579 views

Geometry underlying a comparison of Dieudonné theories

Maybe these hypotheses aren't necessary, but for me $\mathbb G$ will be a smooth formal group of dimension 1 and finite height over a perfect field $k$. There are several presentations of the ...

Eric Peterson

6,308

asked Feb 10, 2017 at 0:36

15 votes

0 answers

517 views

Zariski vs etale torsors over abelian varieties

Question. Let $A$ be an abelian variety (say, over the complex numbers), $G$ an algebraic group, $c$ a class in $H^1_{\rm et}(A, G)$. Denote the multiplication by $N$ map on A by $m_N:A\to A$. Does ...

Piotr Achinger

16.2k

asked Dec 23, 2016 at 15:33

15 votes

0 answers

430 views

Counting abelian varieties over finite fields in a given isogeny class

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...

Will Sawin

149k

asked Sep 11, 2015 at 21:15

15 votes

0 answers

2k views

Why was it so difficult to define the relative de Rham-Witt complex?

In Illusie's original article, the de Rham-Witt complex is defined for a smooth scheme over a perfect characteristic $p$ base $S$, without reference to $S$. Some 25 years later, Langer and Zink ...

Piotr Achinger

16.2k

asked Dec 16, 2013 at 9:02

15 votes

0 answers

403 views

Does every Abelian variety have a finite resolution by Jacobians?

One knows that every Abelian variety is a quotient of a Jacobian. Does every Abelian variety have a finite resolution by Jacobians?

user19475

asked Feb 12, 2012 at 17:22

15 votes

0 answers

779 views

Lifting varieties from char. $p$ to char. 0 after alterations

The question is related to this MO question: Lifting varieties to characteristic zero. Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an ...

shenghao

4,265

asked May 17, 2011 at 13:11

14 votes

0 answers

1k views

A slick proof (?) of Zariski-Nagata purity in characteristic $p$

I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...

Lisa S.

2,663

asked Sep 28, 2017 at 18:00

14 votes

0 answers

562 views

Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s. In the comments of the question, I was directed to the paper http://arxiv.org/abs/...

Simon Rose

6,290

asked Dec 1, 2014 at 12:48

14 votes

0 answers

1k views

Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...

John Binder

1,453

asked Apr 17, 2014 at 20:52

13 votes

0 answers

749 views

Rings whose Frobenius is flat

Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$. I am interested in rings for which $F_R$ is flat (hence ...

anon1432

asked Aug 25, 2020 at 18:46

13 votes

0 answers

348 views

An example of curves with the same Jacobian, but different Jacobian automorphism groups (wrt their respective canonical principal polarizations)?

I am trying to understand examples of differing curves with the same Jacobian, and the quirks of the Jacobian. Here is my question: What is an example where $Aut(Jac(X, a))$ and $Aut(Jac(X', a'))$ ...

Catherine Ray

3,539

asked Jun 27, 2018 at 20:59

13 votes

0 answers

943 views

Beilinson-Bernstein localization in positive characteristic

This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...

Chuck Hague

3,637

asked Feb 10, 2010 at 17:41

12 votes

0 answers

593 views

What is known about abelian varieties with several principal polarizations?

Let ``the simple case" be when the polarized abelian variety does not break up into a product of polarized abelian varieties. I am trying to get an idea of what is known about abelian varieties with ...

Catherine Ray

3,539

asked Jul 26, 2018 at 1:53

12 votes

0 answers

729 views

Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)

Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$. I have seen another post on ...

Catherine Ray

3,539

asked Mar 23, 2016 at 1:12

12 votes

0 answers

273 views

Are Hecke eigenvalues on the cohomology of the Newton polygon strata automorphic?

Fix a genus $g$, a prime $p$, and a Newton polygon $\Delta$ of an abelian variety of genus $g$. Let $\mathcal A_{g, \overline{\mathbb F}_p, \Delta}$ be the moduli stack of abelian varieties of genus $...

Will Sawin

149k

asked Oct 31, 2015 at 20:44

12 votes

0 answers

464 views

Are automorphisms of abelian varieties detected by the formal group?

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$. Assume $k$ has characteristic $p$ and denote by $A(p)$ the $p$-divisible group of dimension $g$ associated with ...

Lennart Meier

12.1k

asked Jan 3, 2014 at 16:27

12 votes

0 answers

433 views

Average ranks of abelian surfaces

Most people nowadays believe that over a fixed global field, $50$% of the elliptic curves have $0$ rank, $50$% have rank $1$, and $0$% have higher rank. A significant advance in this direction has ...

Vesselin Dimitrov

13.8k

asked Feb 5, 2013 at 23:38

12 votes

0 answers

716 views

Lifting abelian varieties in (the closed fiber of) a fixed Neron model

Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...

B. Cais

1,609

asked Mar 19, 2010 at 16:00

11 votes

0 answers

374 views

Example of abelian variety over finite field which doesn't lift

What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given. Note that ...

Daniel Loughran

21.4k

asked Sep 12, 2023 at 16:27

11 votes

0 answers

420 views

Good reduction of finite etale covers of abelian varieties

Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$. Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale ...

Pat

513

asked Feb 11, 2020 at 15:05

11 votes

0 answers

382 views

What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?

Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...

Vesselin Dimitrov

13.8k

asked Jul 2, 2018 at 19:13

11 votes

0 answers

310 views

Surfaces with $q=2$ and generically finite Albanese map

I have a family of surfaces of general type $S$ with $q(S)=2$, and such that the Albanese map $$\alpha \colon S \longrightarrow A:=\mathrm{Alb}(S)$$ is generically finite of degree $n$. By a result of ...

Francesco Polizzi

66.3k

asked Aug 22, 2016 at 13:36

11 votes

0 answers

524 views

Singular curve on an abelian surface

Let $C_2$ be a smooth genus $2$ curve and $J(C_2)$ its Jacobian. It is well known that the blow-up of $J(C_2)$ at the origin $o$ is isomorphic to the second symmetric product $\textrm{Sym}^2(C_2)$, ...

Francesco Polizzi

66.3k

asked Jun 16, 2016 at 17:39

11 votes

0 answers

491 views

Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial? This follows from the conjecture that the maximal (pro-)solvable ...

Pablo

11.3k

asked Mar 1, 2015 at 0:28

11 votes

0 answers

351 views

Purity for abelian schemes up to $p$-isogenies

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the ...

Simon Pepin Lehalleur

1,606

asked Oct 21, 2013 at 15:40

11 votes

0 answers

576 views

What's known about the mod 2 reduction of the level l Jacobi modular equation?

Motivation: Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...

paul Monsky

5,422

asked Jan 21, 2011 at 18:08

11 votes

0 answers

1k views

Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?

It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...

bhwang

1,764

asked Nov 2, 2010 at 19:43

10 votes

0 answers

371 views

How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?

$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...

user509184

1,335

asked Jan 7 at 0:39

10 votes

0 answers

438 views

Boundary of Siegel modular variety

The moduli space of curves has a compactification whose boundary can be understood as the product of moduli spaces of curves of lower genus. Therefore (perhaps naively) one might hope that there ...

curious math guy

4,418

asked Jul 29, 2020 at 3:37

10 votes

0 answers

294 views

Relation between Faltings height and height on moduli space

Let $E$ be an elliptic curve over a number field $K$. The difference between the semistable Faltings height $h_F(E)$ of $E$ and the height $h(j_E)$ of the $j$-invariant of $E$ can be bounded in terms ...

R. Alexander

101

asked Dec 3, 2018 at 9:49

10 votes

0 answers

541 views

Is the compositum of all quadratic extensions of the rationals an ample field?

Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$ Is there a (geometrically irreducible) smooth variety $V/\mathbb{...

Pablo

11.3k

asked Aug 3, 2015 at 9:53

10 votes

0 answers

516 views

About the Bloch conjecture on entire curves

The Bloch conjecture states the following: Bloch's conjecture. Let $X$ be a compact complex Kähler variety such that the irregularity $q = h^0(X,\Omega^1_X)$ is larger than the dimension $n = \dim X$....

diverietti

7,902

asked Oct 24, 2012 at 16:21

10 votes

0 answers

323 views

The mod 3 reduction of some powers of delta

Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix k>0 and ...

paul Monsky

5,422

asked Sep 26, 2012 at 15:22

10 votes

0 answers

492 views

Obstructions to deforming an abelian variety

Are abelian varieties unobstructed? That is, given an abelian scheme $X \to \mathrm{Spec}R $ for $R$ a local artinian ring and $\mathrm{Spec} R \to \mathrm{Spec} S $ a nilpotent thickening, can we ...

Akhil Mathew

25.6k

asked Jun 15, 2012 at 2:32

10 votes

0 answers

270 views

Mod m versions of the toric part of Tate modules

Let $A$ be a polarized abelian variety over a local field $K$ with residue characteristic $p$. In the course of proving that a polarized abelian variety $A/K$ has semi-stable reduction iff for all $\...

Greicius

101

asked May 16, 2011 at 13:36

9 votes

0 answers

439 views

Uncountably many non-isomorphic Tate modules

Do there exist uncountably many abelian surfaces with good reduction over $\mathbb{Q}_p$ with pairwise non-isomorphic rational $p$-adic Tate modules? If we took $l$-adic Tate modules there would be ...

user158636

asked Aug 7, 2020 at 15:20

9 votes

0 answers

383 views

Does bounded-degree base extension yield Zariski-dense Mordell-Weil group?

If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property? For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an ...

Michael Zieve

6,397

asked Sep 3, 2015 at 17:18

9 votes

0 answers

582 views

Kernels and cokernels for morphisms of abelian schemes up to isogenies

For $S$ a noetherian scheme, let $\mathcal{A}(S)$ be the additive category of abelian schemes over $S$ and $\mathcal{A}_{\mathbb{Q}}(S)$ be the category of abelian schemes up to isogenies, i.e. ...

Simon Pepin Lehalleur

1,606

asked Oct 24, 2013 at 15:06

9 votes

0 answers

560 views

Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation

Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...

paul Monsky

5,422

asked Oct 13, 2011 at 23:02

9 votes

0 answers

1k views

Ample vector bundles, $H^1=0$ and global generation in characteristic $p$

This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...

Frank

2,976

asked Jul 8, 2011 at 17:05

8 votes

0 answers

261 views

Simultaneous rank jumping of elliptic curves over number fields

Here's something fun that I've been wondering about for a while out of curiosity. It has to do with the rank $\mathrm{rk}(E(K))$ of an elliptic curve $E$ over a number field $K$, and especially how it ...

Ariyan Javanpeykar

9,497

asked Jan 15, 2022 at 0:22

8 votes

0 answers

201 views

Monodromy groups that are profinitely dense in Sp(2g,Z)

$\DeclareMathOperator\Sp{Sp}$Assume $g\geq 2$. It is known that there exist finitely generated subgroups of $\Sp(2g,\mathbb{Z})$ of infinite index that surject onto all finite quotients of $\Sp(2g,\...

Gabriele Mondello

161

asked Aug 8, 2021 at 13:24

8 votes

0 answers

300 views

Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?

Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...

Andy Jiang

2,356

asked Mar 26, 2021 at 4:28

8 votes

0 answers

688 views

An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich

Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$. In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map $$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...

Joshua Mundinger

2,368

asked Feb 12, 2021 at 18:29

8 votes

0 answers

196 views

Simple abelian varieties of $\mathrm{GL}_2$ type with positive rank and large dimension

$\DeclareMathOperator\GL{GL}$I would like know if there are known constructions of simple abelian varieties of $\GL_2$ type of arbitrarily large dimension and positive Mordell-Weil rank, whose rank is ...

Maarten Derickx

2,224

asked Jan 18, 2021 at 10:42

8 votes

0 answers

471 views

Sheaf whose singular support is not Lagrangian

For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...

John Pardon

18.7k

asked Sep 18, 2017 at 14:58

8 votes

0 answers

636 views

Weil pairing and Tate module for $p$-torsion in characteristic $p$

Let $A/\mathbf{F}_{p^n}$ be an Abelian variety with $\bar{A} = A \times_{\mathbf{F}_{p^n}} \bar{\mathbf{F}}_{p^n}$. If $\ell \neq p$ is prime, there is a natural isomorphism $$\mathrm{H}^2_{\mathrm{...

user19475

asked Mar 7, 2017 at 14:43

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